3.778 \(\int \left (a+\frac{b}{x^2}\right ) \left (c+\frac{d}{x^2}\right )^{3/2} x \, dx\)

Optimal. Leaf size=110 \[ -\frac{\left (c+\frac{d}{x^2}\right )^{3/2} (3 a d+2 b c)}{6 c}-\frac{1}{2} \sqrt{c+\frac{d}{x^2}} (3 a d+2 b c)+\frac{1}{2} \sqrt{c} (3 a d+2 b c) \tanh ^{-1}\left (\frac{\sqrt{c+\frac{d}{x^2}}}{\sqrt{c}}\right )+\frac{a x^2 \left (c+\frac{d}{x^2}\right )^{5/2}}{2 c} \]

[Out]

-((2*b*c + 3*a*d)*Sqrt[c + d/x^2])/2 - ((2*b*c + 3*a*d)*(c + d/x^2)^(3/2))/(6*c)
+ (a*(c + d/x^2)^(5/2)*x^2)/(2*c) + (Sqrt[c]*(2*b*c + 3*a*d)*ArcTanh[Sqrt[c + d
/x^2]/Sqrt[c]])/2

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Rubi [A]  time = 0.224199, antiderivative size = 110, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ -\frac{\left (c+\frac{d}{x^2}\right )^{3/2} (3 a d+2 b c)}{6 c}-\frac{1}{2} \sqrt{c+\frac{d}{x^2}} (3 a d+2 b c)+\frac{1}{2} \sqrt{c} (3 a d+2 b c) \tanh ^{-1}\left (\frac{\sqrt{c+\frac{d}{x^2}}}{\sqrt{c}}\right )+\frac{a x^2 \left (c+\frac{d}{x^2}\right )^{5/2}}{2 c} \]

Antiderivative was successfully verified.

[In]  Int[(a + b/x^2)*(c + d/x^2)^(3/2)*x,x]

[Out]

-((2*b*c + 3*a*d)*Sqrt[c + d/x^2])/2 - ((2*b*c + 3*a*d)*(c + d/x^2)^(3/2))/(6*c)
+ (a*(c + d/x^2)^(5/2)*x^2)/(2*c) + (Sqrt[c]*(2*b*c + 3*a*d)*ArcTanh[Sqrt[c + d
/x^2]/Sqrt[c]])/2

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Rubi in Sympy [A]  time = 16.9551, size = 94, normalized size = 0.85 \[ \frac{a x^{2} \left (c + \frac{d}{x^{2}}\right )^{\frac{5}{2}}}{2 c} + \sqrt{c} \left (\frac{3 a d}{2} + b c\right ) \operatorname{atanh}{\left (\frac{\sqrt{c + \frac{d}{x^{2}}}}{\sqrt{c}} \right )} - \sqrt{c + \frac{d}{x^{2}}} \left (\frac{3 a d}{2} + b c\right ) - \frac{\left (c + \frac{d}{x^{2}}\right )^{\frac{3}{2}} \left (\frac{3 a d}{2} + b c\right )}{3 c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((a+b/x**2)*(c+d/x**2)**(3/2)*x,x)

[Out]

a*x**2*(c + d/x**2)**(5/2)/(2*c) + sqrt(c)*(3*a*d/2 + b*c)*atanh(sqrt(c + d/x**2
)/sqrt(c)) - sqrt(c + d/x**2)*(3*a*d/2 + b*c) - (c + d/x**2)**(3/2)*(3*a*d/2 + b
*c)/(3*c)

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Mathematica [A]  time = 0.146106, size = 109, normalized size = 0.99 \[ \frac{\sqrt{c+\frac{d}{x^2}} \left (\sqrt{c x^2+d} \left (3 a c x^4-6 a d x^2-2 b \left (4 c x^2+d\right )\right )+3 \sqrt{c} x^3 (3 a d+2 b c) \log \left (\sqrt{c} \sqrt{c x^2+d}+c x\right )\right )}{6 x^2 \sqrt{c x^2+d}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b/x^2)*(c + d/x^2)^(3/2)*x,x]

[Out]

(Sqrt[c + d/x^2]*(Sqrt[d + c*x^2]*(-6*a*d*x^2 + 3*a*c*x^4 - 2*b*(d + 4*c*x^2)) +
3*Sqrt[c]*(2*b*c + 3*a*d)*x^3*Log[c*x + Sqrt[c]*Sqrt[d + c*x^2]]))/(6*x^2*Sqrt[
d + c*x^2])

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Maple [B]  time = 0.019, size = 203, normalized size = 1.9 \[{\frac{1}{6\,{d}^{2}} \left ({\frac{c{x}^{2}+d}{{x}^{2}}} \right ) ^{{\frac{3}{2}}} \left ( 6\,b{c}^{3/2}\ln \left ( \sqrt{c}x+\sqrt{c{x}^{2}+d} \right ){x}^{3}{d}^{2}+6\,ac{x}^{4} \left ( c{x}^{2}+d \right ) ^{3/2}d+4\,b{c}^{2}{x}^{4} \left ( c{x}^{2}+d \right ) ^{3/2}+9\,a\sqrt{c}{d}^{3}\ln \left ( \sqrt{c}x+\sqrt{c{x}^{2}+d} \right ){x}^{3}-6\,a \left ( c{x}^{2}+d \right ) ^{5/2}{x}^{2}d-4\,bc \left ( c{x}^{2}+d \right ) ^{5/2}{x}^{2}+9\,ac{x}^{4}\sqrt{c{x}^{2}+d}{d}^{2}+6\,b{c}^{2}{x}^{4}\sqrt{c{x}^{2}+d}d-2\,b \left ( c{x}^{2}+d \right ) ^{5/2}d \right ) \left ( c{x}^{2}+d \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b/x^2)*(c+d/x^2)^(3/2)*x,x)

[Out]

1/6*((c*x^2+d)/x^2)^(3/2)*(6*b*c^(3/2)*ln(c^(1/2)*x+(c*x^2+d)^(1/2))*x^3*d^2+6*a
*c*x^4*(c*x^2+d)^(3/2)*d+4*b*c^2*x^4*(c*x^2+d)^(3/2)+9*a*c^(1/2)*d^3*ln(c^(1/2)*
x+(c*x^2+d)^(1/2))*x^3-6*a*(c*x^2+d)^(5/2)*x^2*d-4*b*c*(c*x^2+d)^(5/2)*x^2+9*a*c
*x^4*(c*x^2+d)^(1/2)*d^2+6*b*c^2*x^4*(c*x^2+d)^(1/2)*d-2*b*(c*x^2+d)^(5/2)*d)/(c
*x^2+d)^(3/2)/d^2

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a + b/x^2)*(c + d/x^2)^(3/2)*x,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.240651, size = 1, normalized size = 0.01 \[ \left [\frac{3 \,{\left (2 \, b c + 3 \, a d\right )} \sqrt{c} x^{2} \log \left (-2 \, c x^{2} - 2 \, \sqrt{c} x^{2} \sqrt{\frac{c x^{2} + d}{x^{2}}} - d\right ) + 2 \,{\left (3 \, a c x^{4} - 2 \,{\left (4 \, b c + 3 \, a d\right )} x^{2} - 2 \, b d\right )} \sqrt{\frac{c x^{2} + d}{x^{2}}}}{12 \, x^{2}}, \frac{3 \,{\left (2 \, b c + 3 \, a d\right )} \sqrt{-c} x^{2} \arctan \left (\frac{c}{\sqrt{-c} \sqrt{\frac{c x^{2} + d}{x^{2}}}}\right ) +{\left (3 \, a c x^{4} - 2 \,{\left (4 \, b c + 3 \, a d\right )} x^{2} - 2 \, b d\right )} \sqrt{\frac{c x^{2} + d}{x^{2}}}}{6 \, x^{2}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a + b/x^2)*(c + d/x^2)^(3/2)*x,x, algorithm="fricas")

[Out]

[1/12*(3*(2*b*c + 3*a*d)*sqrt(c)*x^2*log(-2*c*x^2 - 2*sqrt(c)*x^2*sqrt((c*x^2 +
d)/x^2) - d) + 2*(3*a*c*x^4 - 2*(4*b*c + 3*a*d)*x^2 - 2*b*d)*sqrt((c*x^2 + d)/x^
2))/x^2, 1/6*(3*(2*b*c + 3*a*d)*sqrt(-c)*x^2*arctan(c/(sqrt(-c)*sqrt((c*x^2 + d)
/x^2))) + (3*a*c*x^4 - 2*(4*b*c + 3*a*d)*x^2 - 2*b*d)*sqrt((c*x^2 + d)/x^2))/x^2
]

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Sympy [A]  time = 17.7261, size = 187, normalized size = 1.7 \[ \frac{3 a \sqrt{c} d \operatorname{asinh}{\left (\frac{\sqrt{c} x}{\sqrt{d}} \right )}}{2} + \frac{a c \sqrt{d} x \sqrt{\frac{c x^{2}}{d} + 1}}{2} - \frac{a c \sqrt{d} x}{\sqrt{\frac{c x^{2}}{d} + 1}} - \frac{a d^{\frac{3}{2}}}{x \sqrt{\frac{c x^{2}}{d} + 1}} + b c^{\frac{3}{2}} \operatorname{asinh}{\left (\frac{\sqrt{c} x}{\sqrt{d}} \right )} - \frac{b c^{2} x}{\sqrt{d} \sqrt{\frac{c x^{2}}{d} + 1}} - \frac{b c \sqrt{d}}{x \sqrt{\frac{c x^{2}}{d} + 1}} + b d \left (\begin{cases} - \frac{\sqrt{c}}{2 x^{2}} & \text{for}\: d = 0 \\- \frac{\left (c + \frac{d}{x^{2}}\right )^{\frac{3}{2}}}{3 d} & \text{otherwise} \end{cases}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a+b/x**2)*(c+d/x**2)**(3/2)*x,x)

[Out]

3*a*sqrt(c)*d*asinh(sqrt(c)*x/sqrt(d))/2 + a*c*sqrt(d)*x*sqrt(c*x**2/d + 1)/2 -
a*c*sqrt(d)*x/sqrt(c*x**2/d + 1) - a*d**(3/2)/(x*sqrt(c*x**2/d + 1)) + b*c**(3/2
)*asinh(sqrt(c)*x/sqrt(d)) - b*c**2*x/(sqrt(d)*sqrt(c*x**2/d + 1)) - b*c*sqrt(d)
/(x*sqrt(c*x**2/d + 1)) + b*d*Piecewise((-sqrt(c)/(2*x**2), Eq(d, 0)), (-(c + d/
x**2)**(3/2)/(3*d), True))

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GIAC/XCAS [A]  time = 0.362733, size = 304, normalized size = 2.76 \[ \frac{1}{2} \, \sqrt{c x^{2} + d} a c x{\rm sign}\left (x\right ) - \frac{1}{4} \,{\left (2 \, b c^{\frac{3}{2}}{\rm sign}\left (x\right ) + 3 \, a \sqrt{c} d{\rm sign}\left (x\right )\right )}{\rm ln}\left ({\left (\sqrt{c} x - \sqrt{c x^{2} + d}\right )}^{2}\right ) + \frac{2 \,{\left (6 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + d}\right )}^{4} b c^{\frac{3}{2}} d{\rm sign}\left (x\right ) + 3 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + d}\right )}^{4} a \sqrt{c} d^{2}{\rm sign}\left (x\right ) - 6 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + d}\right )}^{2} b c^{\frac{3}{2}} d^{2}{\rm sign}\left (x\right ) - 6 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + d}\right )}^{2} a \sqrt{c} d^{3}{\rm sign}\left (x\right ) + 4 \, b c^{\frac{3}{2}} d^{3}{\rm sign}\left (x\right ) + 3 \, a \sqrt{c} d^{4}{\rm sign}\left (x\right )\right )}}{3 \,{\left ({\left (\sqrt{c} x - \sqrt{c x^{2} + d}\right )}^{2} - d\right )}^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a + b/x^2)*(c + d/x^2)^(3/2)*x,x, algorithm="giac")

[Out]

1/2*sqrt(c*x^2 + d)*a*c*x*sign(x) - 1/4*(2*b*c^(3/2)*sign(x) + 3*a*sqrt(c)*d*sig
n(x))*ln((sqrt(c)*x - sqrt(c*x^2 + d))^2) + 2/3*(6*(sqrt(c)*x - sqrt(c*x^2 + d))
^4*b*c^(3/2)*d*sign(x) + 3*(sqrt(c)*x - sqrt(c*x^2 + d))^4*a*sqrt(c)*d^2*sign(x)
- 6*(sqrt(c)*x - sqrt(c*x^2 + d))^2*b*c^(3/2)*d^2*sign(x) - 6*(sqrt(c)*x - sqrt
(c*x^2 + d))^2*a*sqrt(c)*d^3*sign(x) + 4*b*c^(3/2)*d^3*sign(x) + 3*a*sqrt(c)*d^4
*sign(x))/((sqrt(c)*x - sqrt(c*x^2 + d))^2 - d)^3